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Asymptotically unitary equivalence and asymptotically inner automorphisms
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 6, December 2009
- pp. 1589-1677
- 10.1353/ajm.0.0086
- Article
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Let $C$ be a unital AH-algebra and let $A$ be a unital separable
simple $C^*$-algebra with tracial rank zero. Suppose that $\phi_1, \phi_2\colon \
C\to A$ are two unital monomorphisms. We show that there is a
continuous path of unitaries $\{u_t\colon \ t\in [0, \infty)\}$ of $A$
such that
$$
\lim_{t\to\infty}u_t^*\phi_1(a)u_t=\phi_2(a)\quad\hbox{\rm for all}\quad a\in C
$$
if and only if $[\phi_1]=[\phi_2]$ in $KK(C,A),$ $\tau\circ
\phi_1=\tau\circ \phi_2$ for all $\tau\in T(A)$ and the rotation
map ${\tilde\eta}_{\phi_1,\phi_2}$ associated with $\phi_1$ and
$\phi_2$ is zero. In particular, an automorphism $\alpha$ on a
unital separable simple $C^*$-algebra $A$ in ${\cal N}$ with tracial rank
zero is asymptotically inner if and only if
$$
[\alpha]=[{\rm id}_A]\quad{\rm in}\quad KK(A,A)
$$
and the rotation map ${\tilde\eta}_{\phi_1, \phi_2}$ is zero.
Let $A$ be a unital AH-algebra (not necessarily simple) and let
$\alpha\in Aut(A)$ be an automorphism. As an application, we show
that the associated crossed product $A\rtimes_{\alpha}{\Bbb Z}$ can be
embedded into a unital simple AF-algebra if and only if $A$ admits
a strictly positive $\alpha$-invariant tracial state.